Every classical knot \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) is homotopic to the unknot.

Generically, the homotopy \(H \colon \mathbb{S}^{1} \times [0, 1] \rightarrow \mathbb{S}^{3}\) is a sequence of isotopies and crossing changes.

Knowing how the (immersed) surfaces in a 4-manifold \(M^{4}\) interact can tell us a lot about the topology of \(M\)

• \(\leadsto\) intersection form \(\pi_{2}(M) \otimes \pi_2(M) \rightarrow \mathbb{Z}\) of \(M\)

Given non-closed 4-manifold \(X^{4}\) with 3-manifold boundary \(\partial X = Y\); classical knot \(k \colon \mathbb{S}^{3} \hookrightarrow Y^{3}\),

• we can investigate surfaces \(\Sigma^{2} \subset X^{4}\) with boundary \(\partial \Sigma = k\)
• \(\leadsto\) 4-dimensional invariants of \(k\)

Related to higher dimensions:

In a 5-dimensional cobordism, the attaching spheres of the 3-handles are 2-knots in 4-manifolds.

(Smale 1958): For every smoothly knotted 2-sphere \(K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\), there is a regular homotopy starting at the embedding \(K\) and ending at the unknot.

This “movie of movies” is a regular homotopy from a non-trivially knotted 2-sphere (left) to the unknotted sphere in \(\mathbb{S}^{4}\) (right)

Define the Casson-Whitney number \[ \mathop{\mathrm{u_{\mathrm{CW}}}}(K) \text{ of } K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4} \] as the minimal number of finger moves in a regular homotopy \(K \leadsto\) unknot.

\(\pi K = \pi_{1}(\mathbb{S}^{4} - K) \twoheadrightarrow H_{4}(\mathbb{S}^{4} - K) \cong \mathbb{Z}\cong \langle t \rangle\)

Study the homology of the associated infinite cyclic cover:

\[ \langle t \rangle \curvearrowright \widetilde{(\mathbb{S}^{4} - K)}_{\mathbb{Z}} \rightarrow \mathbb{S}^{4} - K \]

The Alexander module is the following \(\mathbb{Z}[t, t^{1}]\)-module:

\[ \begin{align} H_{1}(\widetilde{(\mathbb{S}^{4} - K)}_{\mathbb{Z}}) & \cong [\pi K, \pi K] \left/ [[\pi K, \pi K], [\pi K, \pi K]] \right. \\ & \cong \pi K^{(1)} \left/ \pi K^{(2)} \right. \end{align} \]

\[ \pi_{1}(\mathbb{S}^{4} - K') \cong { \pi_{1}(\mathbb{S}^{4} - K) } \; \left/ \; { \langle \langle [ {\color{purple} w}^{{\color{purple} {-1}}} {\color{red} a} {\color{purple} w}, {\color{green} b}] \rangle \rangle } \right. \]

Slogan: Finger moves can make a pair of meridians commute.

\[ { \Large \begin{align} \mathop{\mathrm{u_{\mathrm{CW}}}}(K) & \ge \; \substack{\text{minimal number of finger move relations} \\ [w_{i}^{-1} a_{i} w_{i}, a_{i}] \text{ to make } \pi_{1}(\mathbb{S}^{4} - K) \text{ abelian}} \\[1em] & \ge \; \substack{\text{minimal number of relations} \\ w_{i}^{-1} a_{i} w_{i} = a_{i} \text{ to make } \pi_{1}(\mathbb{S}^{4} - K) \text{ abelian}} \\[1em] & \ge \; \substack{\text{minimal size of generating set} \\ \text{of Alexander module of } K} \end{align} } \]

Rim surgery: surface \(F^{2} \subset X^{4}\), pattern knots \({\color{blue} J_{\color{blue} 1}}, {\color{blue} J_{\color{blue} 2}} \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\)

Thm. The Casson-Whitney number \(\mathop{\mathrm{u_{\mathrm{CW}}}}(\tau^{n} k)\) of every twist spin of \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) is a lower bound for the classical unknotting number of the original knot \(k\).

Special case of (Scharlemann 1985): \(k_{1}, k_{2}\) classical knots with nontrivial determinant, then the classical unknotting number \(u(k_{1} \mathop \#k_{2}) \ge 2\).

Suciu’s infinite family of ribbon 2-knots \(R_{l} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\)

For every \(k \in \mathbb{N}\), \(\pi_{1}(\mathbb{S}^{4} - R_{k})\) is the trefoil knot group

• The fusion number is an upper bound on the Casson-Whitney number of ribbon 2-knots.
• The Casson-Whitney number and the stabilization number are different: \(\exists K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\) with \(\mathop{\mathrm{u_{\textrm{st}}}}(K) \neq \mathop{\mathrm{u_{\mathrm{CW}}}}(K)\)
• \(\mathop{\mathrm{u_{\textrm{st}}}}(K) \le \mathop{\mathrm{u_{\mathrm{CW}}}}(K) + 1\)
• \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K) = 1\) implies \(\mathop{\mathrm{u_{\textrm{st}}}}(K) = 1\)

I found a 4-manifold \(X^{4}\) and non-local knots in \(\partial X\) which are

• topologically shallow slice,
• but smoothly deep slice.

(Freedman et al. 2010)’s attempt at proving a proposed smooth homotopy 4-ball \(\mathcal{B}\) (= smooth contractible compact 4-manifold with \(\mathbb{S}^{3}\) boundary) is exotic:

• Find knot \(K \subset \mathbb{S}^{3} = \partial \mathcal{B}\) that bounds a smooth slice disk in \(\mathcal{B}\),
• but \(K\) does not bound any such disk that is contained in a collar \(\mathbb{S}^{3} \times [0, 1]\) of the boundary of \(\mathcal{B}\).

• \(X^{4}\) is exotic if there exists a smooth 4-manifold \(X'\) such that
  • \(X\) is homeomorphic to \(X'\),
  • but \(X\) is not diffeomorphic to \(X'\)
• (Manolescu, Marengon, and Piccirillo 2020):
  • \(X = \#^{3} \mathbb{CP}^{2} \#^{20} \overline{\mathbb{CP}^{2}}\)
  • \(X' = K3 \# \overline{\mathbb{CP}^{2}}\)
  • \(K =\) right handed trefoil
  • 4-manifolds \(X\) and \(X'\) are homeomorphic
  • \(K\) is smoothly \(H\)-slice in \(X\)
  • but \(K\) not smoothly \(H\)-slice in \(X'\).

I found an infinite family of topologically shallow slice and smoothly deep slice knots in the knot trace \(X_{-1}(T_{2, -3})\):

\((p, 1)\)-cable \(J_{p, 1}\) of the knots \[ J = \textrm{Wh}^{(3)}_{+}( \textrm{Wh}_{+}(K^{\ast}, +1), 0 ) \subset \mathbb{S}^{3}_{-1}(T_{2, -3}) = P \]

\[ J = \textrm{Wh}_{+, 1\textrm{st}}^{(3)}(\mathop{\mathrm{BD}}(K^{\ast}, 0), 0), \]

(R., 2021): There exists a 2-component link \(J\) in the boundary \(Y = \partial X\) of the 2-handlebody \(X = X_{-1}(T_{2, -3})\) with the following properties:

• \(J\) is not a local link;
• \(J\) is link homotopic to the split unlink in the boundary;
• \(J\) is a boundary link;
• \(J\) is Brunnian
• \(J\) bounds smooth strong slice disks in the 2-handlebody \(X\);
• \(J\) is topologically shallow strong slice in \(Y \times [0, 1]\);
• \(J\) is not smoothly shallow strong slice in \(Y \times [0,1]\).